Uniaxial polarization analysis of bulk ferromagnets: theory and first experimental results

Based on the continuum theory of micromagnetics, theoretical expressions for the polarization of the scattered neutron beam in uniaxial small-angle neutron scattering have been derived and their predictions tested by analyzing experimental data on a soft magnetic nanocrystalline alloy. The here-presented theoretical framework forms the basis for polarized real-space methods such as spin-echo small-angle neutron scattering, spin-echo modulated small-angle neutron scattering and polarized neutron dark-field contrast imaging.

However, for the scattering of cold (long-wavelength) neutrons along the forward direction -as implemented on a small-angle neutron scattering (SANS) instrument -it has only in recent years become possible to perform neutron polarization analysis (retaining the full two-dimensional scattering information) 'routinely': more specifically, uniaxial (also called longitudinal or one-dimensional) polarization analysis, where the polarization of the scattered neutrons is analyzed along the direction of the initial polarization (Moon et al., 1969). Clearly, this progress is due to the development of efficient 3 He spin filters (e.g. Batz et al., 2005;Petoukhov et al., 2006;Okudaira et al., 2020), which, in contrast to e.g. singlecrystal analyzers, can be used over a rather broad wavelength range and cover a large detector acceptance angle. Note also that Niketic et al. (2015) and Quan et al. (2019a,b) report on the development of a novel neutron spin filter, based on the strong spin dependence of the neutron scattering on protons. For the combination of uniaxial polarization analysis with SANS, the term POLARIS has been coined (Wiedenmann, 2005). In contrast to CRYOPAD, which generally demands the sample to be in a zero magnetic field environment, POLARIS allows for the application of large magnetic fields.
The POLARIS method has been successfully employed for studying the superparamagnetic response of concentrated ferrofluids (Wiedenmann, 2005), proton domains in deuterated solutions (van den Brandt et al., 2006;Aswal et al., 2008;Noda et al., 2016), the multiferroic properties of HoMn 3 single crystals (Ueland et al., 2010), the role of nanoscale heterogeneities for the magnetostriction of Fe-Ga alloys Laver et al., 2010), local weak ferromagnetism in BiFeO 3 (Ramazanoglu et al., 2011), nanometre-sized magnetic domains and coherent magnetization reversal in an exchange-bias system (Dufour et al., 2011), precipitates in Heusler-based alloys (Benacchio et al., 2019), the magnetic microstructure of nanoscaled bulk magnets (Honecker et al., 2010;Michels et al., 2012), the internal spin structure of nanoparticles (Krycka et al., 2010Grutter et al., 2017;Orue et al., 2018;Bender, Fork et al., 2018;Oberdick et al., 2018;Ijiri et al., 2019;Bender et al., 2019;Honecker et al., 2020), and Invar alloys (Stewart et al., 2019). Polarization analysis further makes it possible to reveal the direction of the magnetic anisotropy in single-crystalline spin systems, e.g. an easy plane versus an easy axis anisotropy or the confinement of the propagation vector along certain crystallographic directions in chiral and other exotic magnets (Takagi et al., 2018;. In all of the above-mentioned studies, the spinresolved SANS cross sections were obtained and analyzed, but the polarization of the scattered neutrons was not further investigated. Historically, this is the domain of the neutron depolarization technique (see e.g. Halpern & Holstein, 1941;Hughes et al., 1948Hughes et al., , 1949Burgy et al., 1950;Maleev & Ruban, 1970;Rekveldt, 1971Rekveldt, , 1973Drabkin et al., 1972;Maleev & Ruban, 1972;Rosman & Rekveldt, 1991, and references therein), where one measures the change in the polarization of a polarized neutron beam after transmission through a partially magnetized magnetic material. Analysis of the 3 Â 3 depolarization matrix yields information on e.g. the average domain size and the domain magnetization. This type of polarization analysis on a SANS instrument has been termed 'vector analysis of polarization' by Okorokov & Runov (2001). Alternatively, it has been demonstrated that the neutron spinecho technique can resolve magnetic small-angle scattering Rekveldt et al., 2006). Spin-echo small-angle neutron scattering (SESANS) provides information on correlations on a length scale from about 10 nm to 10 mm. In SESANS, the neutron spin precesses in a constant magnetic field and the neutron runtime difference due to sample scattering results in the dephasing of the neutron spins and in a loss of the measured beam polarization. Magnetic scattering can result in neutron spin-flip events that act as an additional optical element reversing the sense of the Larmor precession. The change of the neutron spin due to magnetic scattering can be exploited to study magnetic systems.
More recently, the method of polarized neutron dark-field contrast imaging (DFI) has been introduced for spatially resolved small-angle scattering studies of magnetic microstructures (Valsecchi et al., 2021). First experimental results on a sintered Nd-Fe-B magnet demonstrated not only that darkfield contrast from half-polarized SANS measurements can be observed but also that it becomes possible to separate and retrieve dark-field contrast for all spin-flip and non-spin-flip channels separately. The polarized DFI method has great potential for analyzing real-space magnetic correlations on a macroscopic length scale, well beyond what can be probed with a conventional SANS instrument. Similarly, first measurements of micrometre-sized magnetic correlations have been performed with an alternative neutron precession technique called spin-echo modulated small-angle neutron scattering (SEMSANS) (Li et al., 2021). With this setup, the spin manipulations are performed before the sample so that the measurement is not sensitive to large stray fields (related e.g. to the sample environment), and it even allows studies under beam depolarizing conditions. The polarization analyzer discriminates the polarization parallel to the analyzing direction, such that the scattering cross sections for the opposite neutron spin state are probed at the sample. The two-dimensional neutron polarization modulation observed on the detector is then integrated to yield the one-dimensional correlation function of the system.
For the above techniques, the analysis of magnetic materials is based on performing a neutron-spin analysis deliberately after the sample. The magnetic spin-flip scattering signal of the sample is utilized as a spin flipper to obtain exclusive sensitivity of the signal on local magnetization components. The projected correlation function is modified with the polarization of the scattered neutrons to reflect the additional phase in precession angles due to the spin-flip event.
In this work, we present a micromagnetic SANS theory for the uniaxial polarization of the scattered neutrons of bulk magnetic materials. The approach has recently been employed to analyze SANS cross sections directly in Fourier space (Michels, 2021) and is here extended to include the final polarization, which can be measured with a much higher precision than the individual cross sections (Brown, 2006). The continuum theory of micromagnetics allows one to characterize the large-scale magnetization distribution of polycrystalline magnets, which is determined e.g. by magnetic anisotropy and saturation-magnetization fluctuations, antisymmetric exchange, and dipolar stray fields. Since the validity of micromagnetic theory extends to the micrometre regime, the theoretical framework developed here may also serve as a basis for the above-mentioned polarized neutron techniques (SESANS, DFI, SEMSANS), which provide real-space information on large-scale magnetic correlations. Rekveldt et al. (2006) summarize the relevant expressions which relate the magnetization distribution of the material (obtained from micromagnetics) to the final polarization and the projected correlation function. The derived theoretical expressions for the polarization are tested against experimental SANS data on a soft magnetic nanocrystalline alloy.
The present paper is organized as follows: Section 2 summarizes the elementary equations of polarized neutron scattering, while Section 3 sketches the basic steps and ideas of the micromagnetic SANS theory. These two sections are well documented in the literature and may be skipped by the reader. They are included here merely to achieve a selfcontained presentation. Section 4 gives the final expressions for the polarization of the scattered neutrons, discusses special sector averages and shows the results for the 2 azimuthally averaged saturated state. Section 5 furnishes the details of the polarized SANS experiment and on the investigated sample, while Section 6 presents and discusses the analysis of the experimental results. Section 7 summarizes the main findings of this study. Appendix A features the expressions for the spin-resolved SANS cross sections in terms of the Fourier components of the magnetization, which enter the final expressions for the polarization, while Appendix B showcases some computed examples for the polarization. Fig. 1 depicts a typical uniaxial neutron polarization analysis setup. We consider the most relevant cases where the externally applied magnetic field H 0 , which defines the polarization axis for both the incident and scattered neutrons, is either perpendicular or parallel to the wavevector k 0 of the incident neutron beam. Note that in both scattering geometries H 0 is assumed to be parallel to the e z direction of a Cartesian laboratory coordinate system.

Uniaxial SANS polarization analysis
In a classical picture, the polarization P of a neutron beam containing N spins can be defined as the average over the individual polarizations P j of the neutrons (Schweizer, 2006): where 0 jPj 1. In experimental SANS studies the beam is usually partially polarized along a certain guide-field direction (quantization axis), which we take here as the z direction.
Assuming that the expectation values of the perpendicular polarization components vanish, i.e. P x ¼ P y ¼ 0, and that P z ¼ P, one can then introduce the fractions of neutrons in the spin-up (+) and spin-down (À) states, with p þ þ p À ¼ 1 and p þ À p À ¼ P: Obviously, for an unpolarized beam p þ ¼ p À ¼ 0:5 and P ¼ 0, while P ¼ þ1 (p þ ¼ 1) or P ¼ À1 (p À ¼ 1) for a fully polarized beam. When there is an additional analyzer behind the sample, configured such that it selects only neutrons with spins either parallel or antiparallel to the initial polarization, then one can distinguish four scattering cross sections (scattering processes) (Blume, 1963;Moon et al., 1969;Schweizer, 2006): two that conserve the neutron-spin direction (++ and ÀÀ), called the non-spin-flip cross sections and two cross sections which reverse the neutron spin (+À and À+), called the spin-flip cross sections research papers Figure 1 Sketch of the SANS setup and of the two most often employed scattering geometries in magnetic SANS experiments. (a) Applied magnetic field H 0 perpendicular to the incident neutron beam (k 0 ? H 0 ); (b) k 0 k H 0 . The momentum-transfer or scattering vector q corresponds to the difference between the wavevectors of the incident (k 0 ) and the scattered (k 1 ) neutrons, i.e. q ¼ k 0 À k 1 . Its magnitude for elastic scattering, q ¼ jqj ¼ ð4=Þ sinð Þ, depends on the mean wavelength of the neutrons and on the scattering angle 2 . For a given , sample-todetector distance L SD and distance r D from the centre of the direct beam to a certain pixel element on the detector, the q value can be obtained using q ffi k 0 ðr D =L SD Þ. The symbols 'P', 'F' and 'A' denote, respectively, the polarizer, spin flipper and analyzer, which are optional neutron optical devices. Note that a second flipper after the sample has been omitted here. In spin-resolved SANS (POLARIS) using a 3 He spin filter, the transmission (polarization) direction of the analyzer can be switched by 180 by means of a radiofrequency pulse. SANS is usually implemented as elastic scattering (k 0 ¼ k 1 ¼ 2=), and the component of q along the incident neutron beam [i.e. q x ¼ 0 in (a) and q z ¼ 0 in (b)] is neglected. The angle may be conveniently used in order to describe the angular anisotropy of the recorded scattering pattern on a two-In these expressions, K ¼ 8 3 V À1 b 2 H , where V denotes the scattering volume. b H ¼ 2:70 Â 10 À15 m À1 B ¼ 2:91 Â 10 8 A À1 m À1 is a constant (with B the Bohr magneton) that relates the atomic magnetic moment a to the atomic magnetic scattering length b m , given by (Moon et al., 1969) n ¼ 1:913 denotes the neutron magnetic moment expressed in units of the nuclear magneton, r 0 ¼ 2:818 Â 10 À15 m is the classical radius of the electron and f ðqÞ is the normalized atomic magnetic form factor, which we set to unity, f ffi 1, along the forward direction. The function e N NðqÞ denotes the Fourier transform of the nuclear scattering-length density NðrÞ. The partial SANS cross sections, equations (4) and (5), are written here in terms of the Cartesian components of the Halpern-Johnson vector e Q Q (sometimes also denoted as the magnetic interaction or magnetic scattering vector) (Halpern & Johnson, 1939): whereq q is the unit scattering vector, and e M MðqÞ ¼ f e M M x ðqÞ; e M M y ðqÞ; e M M z ðqÞg represents the Fourier transform of the magnetization vector field MðrÞ ¼ fM x ðrÞ; M y ðrÞ; M z ðrÞg of the sample under study. The three-dimensional Fourier-transform pair of the magnetization is defined as follows: MðrÞ exp Àiq Á r ð Þd 3 r; ð8Þ The Halpern-Johnson vector is a manifestation of the dipolar origin of magnetic neutron scattering, and it emphasizes the fact that only the components of M that are perpendicular to q are relevant for magnetic neutron scattering. We note that different symbols for the Halpern-Johnson vector such as M ? , Q ? , S ? or q, as in the original paper by Halpern & Johnson (1939), can be found in the literature. Likewise, in many textbooks (e.g. Lovesey, 1984;Squires, 2012) e Q Q is defined with a minus sign and normalized by the factor 2 B , which makes it dimensionless. e Q Q is a linear vector function of the components of e M M. Both e Q QðqÞ and e M MðqÞ are in general complex vectors. For k 0 ? H 0 and k 0 k H 0 one finds, respectively (subscripts ? and k refer to the respective scattering geometry, compare Fig. 1), Inserting these expressions into equation (7) Inspection of equations (4) and (5) shows that the transverse components e Q Q x and e Q Q y give rise to spin-flip scattering, while the longitudinal component e Q Q z results in non-spin-flip scattering. Furthermore, if we set ¼ 0 in equation (12), which corresponds to the case that the scattering vector is along the neutron polarization, we see that so that the magnetic scattering along the polarization direction is purely spin flip, and nuclear coherent and magnetic scattering are fully separated in the perpendicular scattering geometry [compare also with equations (4) and (5) for the non-spin-flip and the spin-flip SANS cross sections and with equation (21) for the final polarization]. In the case k 0 k H 0 [equation (13)], spin-flip scattering probes only the transverse magnetization Fourier components e M M x;y , whereas the longitudinal scattering is entirely contained in the non-spin-flip channel, in contrast to the k 0 ? H 0 geometry. We emphasize that nuclear-spin-dependent SANS is not taken into account in this paper, so that the corresponding scattering contributions do not show up in equations (4) and (5).
The total SANS cross section dAE=d can be expressed in terms of the initial spin populations p AE as (Blume, 1963;Moon et al., 1969;Schweizer, 2006) Inserting the above expressions for p þ and p À [equations (2) and (3)] and for the partial SANS cross sections dAE AEAE =d and dAE AEÇ =d [equations (4) and (5)], equation (15) evaluates to which, using P ¼ f0; 0; P z ¼ AEPg, can be rewritten as Since the cross section is a scalar quantity and the polarization is an axial vector (or pseudovector), equation (16) shows that the system under study must itself contain an axial vector. As emphasized by Maleev (2002), examples for such built-in pseudovectors are related to the interaction of a polycrystalline sample with an external magnetic field (inducing an average magnetization directed along the applied field), the existence of a spontaneous magnetization in a ferromagnetic single crystal, the antisymmetric Dzyaloshinskii-Moriya interaction (DMI), mechanical (torsional) deformation or the presence of spin spirals. If on the other hand there is no research papers 572 Artem Malyeyev et al. Uniaxial polarization analysis of bulk ferromagnets preferred axis in the system, then dAE=d is independent of P.
Examples include a collection of randomly oriented noninteracting nuclear (electronic) spins, which describe the general case of nuclear (paramagnetic) scattering at not-toolow temperatures and large applied fields, or a multi-domain ferromagnet with a random distribution of the domains. The same condition -existence of an axial system vector -applies for neutrons to be polarized in the scattering process [compare the last two terms in equation (21) below].
In the domain of magnetic SANS it is customary to denote experiments with a polarized incident beam only, and no spin analysis of the scattered neutrons, with the acronym SANSPOL. The two SANSPOL cross sections dAE þ =d and dAE À =d (also sometimes denoted as the half-polarized SANS cross sections) combine non-spin-flip and spin-flip scattering contributions, according to (p AE ¼ 1) Finally, noting that an unpolarized beam can be viewed as consisting of 50% spin-up and 50% spin-down neutrons [compare equations (2) and (3)], the unpolarized SANS cross section is obtained as [compare equation (15)] The polarization P f of the scattered beam along the direction of the incident neutron polarization P is obtained from the following relation (Blume, 1963;Moon et al., 1969;Schweizer, 2006): The first four terms in the second line on the right-hand side of equation (21) demonstrate that nuclear scattering (to be more precise, the nuclear coherent scattering, the isotopic disorder scattering, and 1/3 of the nuclear-spin-dependent scattering) and scattering due to the longitudinal component e Q Q z of the magnetic scattering vector e Q Q do not reverse the initial polarization, while the two transverse components e Q Q x and e Q Q y give rise to spin-flip scattering. The last two terms in equation (21) do create polarization: these are the familiar nuclear-magnetic interference term ( e N N e Q Q Ã z þ e N N Ã e Q Q z ), which is commonly used to polarize beams, and the chiral term ið e Q Q x e Q Q Ã y À e Q Q Ã x e Q Q y Þ, which is of relevance in inelastic scattering (dynamic chirality) (Okorokov et al., 1981;Maleev, 2002;Grigoriev et al., 2015), in elastic scattering on spiral structures and weak ferromagnets (canted antiferromagnets) (Thoma et al., 2021), or in the presence of the DMI in microstructural-defect-rich magnets Quan et al., 2020). We remind the reader that nuclear-spin-dependent scattering is not taken into account in the expressions for the magnetic SANS cross sections. In the general expression for the polarization of the scattered neutrons, a term iP Â ð e N N e Q Q Ã À e N N Ã e Q QÞ appears (Schweizer, 2006), which is ignored in equation (21). This term rotates the polarization perpendicular to the initial polarization and cannot be observed in the uniaxial setup. We emphasize that in linear neutron polarimetry it is not possible to distinguish between a rotation of the polarization vector and a change of its length (Moon et al., 1969;Maleev, 2002).
From equation (21) it follows that the polarization P f ðqÞ of the scattered neutron beam at momentum-transfer vector q can be expressed as (Maleev et al., 1963;Blume, 1963;Brown, 2006) Note that, for the following analysis, we drop the minus sign in front of the round brackets in equation (23b). For a quantitative analysis of P AE f , a theoretical model for the magnetization Fourier components e M M x;y;z ðqÞ and for e N NðqÞ is required. This will be discussed in the next section. Michels et al. (2016) presented a theory for the magnetic SANS cross section of bulk ferromagnets. The approach, which considers the response of the magnetization to spatially inhomogeneous magnetic anisotropy fields and magnetostatic fields, is based on the continuum theory of micromagnetics, valid close to magnetic saturation, and takes the antisymmetric DMI into account. Here, we recall the basic steps and ideas. The starting point is the static equations of micromagnetics for the bulk, which can be derived from the following expression for the magnetic Gibbs free energy (Brown, 1963;Aharoni, 2000;Kronmü ller & Fä hnle, 2003): where the first term denotes the energy due to the isotropic exchange interaction, the second term is the antisymmetric Dzyaloshinskii-Moriya energy (assuming a cubic symmetry), ! a is the anisotropy energy density, and the last two terms are the energies related, respectively, to the dipolar interaction and the externally applied magnetic field; mðrÞ ¼ MðrÞ=M s ðrÞ (Michels et al., 2016). Variational calculus then leads to the following balance-of-torques equation:

Sketch of micromagnetic SANS theory
which expresses the fact that at static equilibrium the torque on the magnetization MðrÞ due to an effective magnetic field H eff ðrÞ vanishes everywhere inside the material. The effective field is defined as the functional derivative of the ferromagnet's total energy-density functional with respect to the magnetization: where H ex ¼ l 2 M r 2 M is the exchange field, H DMI ¼ Àl D r Â M is due to the DMI, H p ðrÞ is the magnetic anisotropy field, H d ðrÞ is the magnetostatic field and H 0 is a uniform applied magnetic field; r 2 is the Laplace operator and r ¼ @=@x e x þ @=@y e y þ @=@z e z is the gradient operator, where the unit vectors along the Cartesian laboratory axes are, respectively, denoted by e x , e y and e z ( 0 , vacuum permeability). The micromagnetic length scales and are, respectively, related to the magnetostatic interaction and to the DMI. The values for the DMI constant D and for the exchange-stiffness constant A are assumed to be uniform throughout the material, in contrast to the local saturation magnetization M s ðrÞ, which is assumed to depend explicitly on the position r [see also Metlov & Michels (2015)]; Averaging over the directions of the magnetic anisotropy field in the plane perpendicular to the applied field, the magnetic terms for the transverse magnetic field geometry (k 0 ? H 0 , Þsin cos 1 þ p sin 2 À p 2 l 2 D q 2 cos 2 ; The results for the parallel field geometry (k 0 k H 0 , In equations (29) is assumed in the approach-to-saturation regime, which Bersweiler et al. (2022) defined for applied fields where the reduced magnetization M=M 0 > $ 90%. These functions characterize the strength and spatial structure of, respectively, the magnetic anisotropy field H p ðrÞ, with correlation length H , and the local saturation magnetization M s ðrÞ, with correlation length M .

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is a dimensionless function, where is the effective magnetic field [not to be confused with H eff ðrÞ in equation (25)], which depends on the internal magnetic field H i ¼ H 0 À NM 0 (N, demagnetizing factor), on q ¼ jqj and on the exchange length of the field The latter quantity is a measure for the size of inhomogeneously magnetized regions around microstructural lattice defects (Mettus & Michels, 2015). The Fourier coefficient of the longitudinal magnetization e M M z ðqÞ provides information on the spatial variation of the saturation magnetization M s ðrÞ; for instance, j e M M z j 2 / ðÁMÞ 2 in a multiphase magnetic nanocomposite, where ÁM denotes the jump of the magnetization magnitude at internal (particle-matrix) interfaces . Moreover, while the squared Fourier components and the cross terms are even functions of q, it is easily seen that the chiral term À2iðqÞ [equation (32)] is asymmetric in q, which is due to the DMI term: at small fields, when the term / e H H 2 p in the numerator of equation (32) dominates, two extrema parallel and antiparallel to the field axis are observed, whereas at larger fields, when the term / e M M 2 z dominates, additional maxima and minima appear approximately along the detector diagonals (Michels et al., 2016).
By inserting equations (29)-(36) into the SANS cross sections (see Appendix A) and by specifying particular models for the nuclear scattering function e N N 2 ðq nuc Þ, the longitudinal magnetic Fourier component e M M 2 z ðq M Þ and the Fourier coefficient of the magnetic anisotropy field e H H 2 p ðq H Þ, one obtains P AE as a function of the magnitude and orientation of the scattering vector q, the applied magnetic field H 0 , the magnetic interaction parameters (A, D, M 0 , ÁM, H p , M , H ) and microstructural quantities (particle-size distribution, crystallographic texture etc.). We emphasize that the expressions (29) where the amplitudes A 2 M and A 2 H (both in units of A 2 nm À2 ) are, respectively, related to the mean-square magnetization fluctuation and anisotropy-field variation. Of course, other scattering functions such as the form factor of a sphere and various structure-factor models (e.g. a Percus-Yevick hardsphere structure factor) can be straightforwardly implemented (Mettus & Michels, 2015). The characteristic structure sizes of e M M 2 z and e H H 2 p are generally different. We remind the reader that these are related, respectively, to the spatial extent of regions with uniform saturation magnetization ( M ) and magnetic anisotropy field ( H ). Measurement of the magnetic-fielddependent Guinier radius provides a means to determine these correlation lengths as well as the exchange-stiffness constant A . A simple example where M ¼ H is a collection of homogeneous and defect-free magnetic nanoparticles in a magnetic and homogeneous matrix. If, on the other hand, atomically sharp grain boundaries are introduced into such particles, then the direction of the magnetic anisotropy field changes due to the changing set of crystallographic directions at the intraparticle interfaces, but the value of M s may remain the same, so that H < M .  showed, assuming H ¼ M and using the sphere form factor for both e M M 2 z and e H H 2 p , that it is the ratio of H p =ÁM (related to the amplitudes A H and A M ) that determines the angular anisotropy and the asymptotic powerlaw dependence of dAE=d as well as the characteristic decay length of spin-misalignment fluctuations. The ratio of nuclear to longitudinal magnetic scattering is denoted by , which for the general case (that the nuclear correlation length nuc is different from M ) is a function of q. Here, we do not specify a particular nuc and assume this characteristic size to be contained in ðqÞ.

Polarization of scattered beam
When the expressions for the elastic differential spin-flip and SANSPOL cross sections dAE AEÇ =d and dAE AE =d [equations (66)-(70)] are inserted into equations (23a) and (23b), and use is made of the expressions for the magnetization Fourier components [equations (29)-(36)], one obtains, respectively, for the transverse and longitudinal scattering geometry The functions h 1 ðqÞ, h 2 ðqÞ, g 1 ðqÞ and g 2 ðqÞ are independent of the incident neutron beam polarization and are defined as þ j e M M z j 2 sin 2 cos 2 À CT yz sin cos 3 ; ð45Þ g 1 ðqÞ ¼ j e M M x j 2 sin 2 þ j e M M y j 2 cos 2 À CT xy sin cos ; ð47Þ At complete magnetic saturation, when MðrÞ ¼ f0; 0; M s ðrÞg, these expressions reduce to h sat 1 ðqÞ ¼ j e M M s j 2 sin 2 cos 2 ; ð49Þ g sat 1 ðqÞ ¼ 0; ð51Þ where e M M s ðqÞ is the Fourier transform of M s ðrÞ. As can be seen from equations (44a) and (44b), the difference between P þ f and P À f resides, for k 0 ? H 0 , in the nuclear-magnetic interference terms / e N N e M M z and / e N N e M M y , and in ðqÞ, while for k 0 k H 0 the two polarizations differ only by the term / e N N e M M z [equations (44c) and (44d)]. We also remind the reader that the Fourier coefficients in the above expressions are evaluated in the plane of the detector, which for the perpendicular scattering geometry corresponds to the plane q x ¼ 0 and for the parallel geometry to the plane q z ¼ 0 (compare Fig. 1).
The e N N e M M y contribution to equations (44a) and (44b) requires special consideration. This term is expected to be negligible for a polycrystalline statistically isotropic ferromagnet with vanishing fluctuations of the saturation magnetization. This can be seen by scrutinizing the following expression for the e M M y magnetization Fourier component in the perpendicular scattering geometry, corresponding to the plane q x ¼ 0 (Michels, 2021): where e H H px and e H H py denote the Cartesian components of the Fourier transform of the magnetic anisotropy field. If we assume that the nuclear scattering is isotropic and that e H H px and e H H py vary randomly in the plane perpendicular to the field, then the corresponding averages over the direction of the anisotropy field vanish. The only remaining term in the e N N e M M y contribution is then (q y =q ¼ sin ; q z =q ¼ cos ) Þsin cos 1 þ p sin 2 À p 2 l 2 D q 2 cos 2 ; ð54Þ where we have furthermore assumed that both e N NðqÞ and e M M z ðqÞ are real valued, as is done throughout this paper. Note that equation (54) still needs to be multiplied with the term sin cos in order to obtain the corresponding contribution to P AE f? [compare equations (44a) and (44b)]. For homogeneous single-phase materials with M s ¼ constant, the e N N e M M y contribution is expected to be negligible, and we are not aware that this has been reported experimentally. However, for materials exhibiting strong spatial nanoscale variations in the saturation magnetization, i.e. M s ¼ M s ðrÞ, such as magnetic nanocomposites or porous ferromagnets, it should be possible to observe this scattering contribution in polarized SANS experiments.

Sector averages
Carrying out a 2 azimuthal average of P AE f ðqÞ, which are maps with numbers varying between AE1, may result in a significant loss of information (compare e.g. Figs. 10 and 11 below). It is therefore often advantageous to consider cuts of P AE f along certain directions in q space. This might also be of relevance for other spin-manipulating techniques such as SEMSANS, which is a one-dimensional technique that only measures correlations in the encoding direction (Li et al., 2021). However, one has to keep in mind that SEMSANS is a real-space technique that (similar to SESANS and DFI) essentially measures the cosine Fourier transform of the cross section. Nevertheless, the analytical expressions for the magnetization Fourier components can be used in such a transform to obtain information on the magnetic interactions via the projected correlation function.
Sector averages are straightforwardly obtained by evaluating equations (44a)-(44d) [using equations (45)-(52)] for certain angles . For instance, for the perpendicular scattering geometry and for q along the vertical direction on the detector ( ¼ 90 ), we obtain [ðq; where [compare equation (29)] research papers At saturation (M x ¼ 0), P AE f? ðq; ¼ 90 Þ ¼ 1, except for the case ¼ 1, where P þ f? ðq; ¼ 90 Þ ¼ À1. We also see that information on the DMI is contained in j e M M x j 2 via the length scale l D . For l D ¼ 0, j e M M x j 2 ¼ ðp 2 =2Þ e H H 2 p . For the perpendicular scattering geometry and q along the horizontal direction ( ¼ 0 ), we obtain where [compare equations (29) and (30)] and [compare equation (32)] At saturation (M x ¼ M y ¼ ¼ 0) and for nonzero nuclear scattering, P AE f? ðq; H H 2 p and P AE f? ðq; ¼ 0 Þ contains information on the transverse spin components.
In the following, we will consider the cases of ¼ constant and ¼ ðqÞ using the experimental data of the soft magnetic Fe-based alloy NANOPERM (Michels et al., 2012). 4.2.1. a = constant. Fig. 2 displays the two-dimensional polarization P AE f? ðqÞ of the scattered neutrons in the saturated state as a function of = constant. The case of constant is very rarely realized in experimental situations, and we consider it here only as a starting point for our discussion and for the comparison with the experimentally more relevant situation of ¼ ðqÞ. For ! 0 it follows that P AE f? ¼ 1 À 2 cos 2 [Figs. 2(a) and 2(e)], while P þ f? ¼ 1 À 2 sin 2 [ Fig. 2(c)] and P À f? ¼ 1 À 2 sin 2 cos 2 =ð1 þ 3 sin 2 Þ [ Fig. 2(g)] for ¼ 1. When nuclear coherent scattering is dominating ( ! 1), we see that P AE f? both tend to unity, as expected. The corresponding 2 azimuthally averaged functions [equations (62a) and (62b)] are plotted in Fig. 3. One readily verifies that P Plot of P þ f? ðq y ; q z Þ (upper row) and P À f? ðq y ; q z Þ (lower row) in the saturated state for different values of (see insets) [equations (60a) and (60b)].
information when a 2 azimuthal average is carried out [compare e.g. Fig. 2(b)]. 4.2.2. a = a(q). Fig. 4 shows the experimentally determined ratio exp ðqÞ (Michels et al., 2012) of nuclear to magnetic scattering of the two-phase alloy NANOPERM. Within the experimental q range of 0.03 < q < 0.3 nm À1 , these data for exp ðqÞ have been fitted by a power law in 1=q to obtain the functions P AE f? ðqÞ, which are depicted in Fig. 5. The used fit function for exp ðqÞ is exp ðqÞ ¼ 0:14853 q À 0:0264491 q 2 þ 0:00176887 q 3 À 4:95094 Â 10 À5 q 4 þ 5:01767 Â 10 À7 q 5 : This expression will be used in the analysis of the experimental data (see Section 6).

Nonsaturated state
Appendix B features some theoretical results for P AE f? ðqÞ for various combinations of the magnetic interaction parameters [applied magnetic field, ratio of e H H 2 p to e M M 2 z , ðqÞ 6 ¼ 0]. For a statistically isotropic ferromagnet, the two-dimensional distribution of the polarization of the scattered neutrons is isotropic ( independent) for the longitudinal scattering geometry (k 0 k H 0 ), as are the corresponding SANS cross sections. This is in contrast to the P AE f? ðqÞ for the transverse geometry (k 0 ? H 0 ), which are highly anisotropic. In the following, we will use the theoretical expressions for P AE f? ðqÞ to analyze experimental data on the soft magnetic two-phase nanocrystalline alloy NANOPERM.

Experimental details
The polarized neutron experiment was carried out at room temperature at the instrument D22 at the Institut Laue-Langevin, Grenoble, France. Incident neutrons with a mean wavelength of = 8 Å and a wavelength broadening of Á= ¼ 10% (FWHM) were selected by means of a velocity selector. The beam was polarized using a 1.2 m-long remanent Fe-Si supermirror transmission polarizer (m = 3.6), which was installed immediately after the velocity selector. A radiofrequency spin flipper, installed close to the sample position, allowed us to reverse the initial neutron polarization. The external magnetic field (provided by an electromagnet) was applied perpendicular to the wavevector k 0 of the incident neutrons (compare Fig. 1). Measurement of the four partial POLARIS cross sections dAE þþ =d, dAE ÀÀ =d, dAE þÀ =d and dAE Àþ =d was accomplished through a polarized 3 He spinfilter cell, which was installed inside the detector housing, about 1 m away from the sample position. The polarization between polarizer, radiofrequency flipper and 3 He filter was maintained by means of magnetic guide fields of the order of 1 mT. The efficiencies of the polarizer, spin flipper and 3 He analyzer were, respectively, 90, 99 and 87.5%. The scattered neutrons were detected by a multitube detector which consists of 128 Â 128 pixels with a resolution of 8 Â 8 mm. Neutron data reduction, including corrections for background scattering and spin leakage (Wildes, 2006), was performed using the GRASP (Dewhurst, 2021) and BerSANS (Keiderling, 2002;Keiderling et al., 2008) software packages. Black circles: experimental ratio exp ðqÞ of nuclear to magnetic scattering of the two-phase alloy NANOPERM (Michels et al., 2012) (k 0 ? H 0 ; 0 H 0 ¼ 1:27 T; log-log plot). Solid line: power-law fit to parametrize the experimental data [equation (63)]. The fit has been restricted to the interval 0.03 < q < 0.3 nm À1 , but the fit function is displayed for 0.01 < q < 1.0 nm À1 .

Figure 3
Plot of P þ f? and P À f? (see inset) in the saturated state as a function of [equations (62a) and (62b)].
The sample under study was a two-phase magnetic nanocomposite from the NANOPERM family of alloys (Suzuki & Herzer, 2006) with a nominal composition of (Fe 0.985 Co 0.015 ) 90 -Zr 7 B 3 (Suzuki et al., 1994;Ito et al., 2007). The alloy was prepared by melt spinning, followed by a subsequent annealing treatment for 1 h at 883 K, which resulted in the precipitation of body-centered cubic iron nanoparticles in an amorphous magnetic matrix. The average iron particle size of D = 15 AE 2 nm was determined by the analysis of wide-angle X-ray diffraction data. The crystalline particle volume fraction is about 65% and the saturation magnetization of the alloy amounts to 0 M 0 ¼ 1:64 T. The exchange-stiffness constant A ¼ ð4:7 AE 0:9Þ Â 10 À12 J m À1 has previously been determined by the analysis of the field-dependent unpolarized SANS cross section . For the SANS experiments, several circular discs with a diameter of 10 mm and a thickness of about 20 mm were stacked and mounted on a Cd aperture [for further details see Michels et al. (2012) and ].

Experimental results and discussion
The two-dimensional experimental distribution of the polarization of NANOPERM is depicted in Figs  In agreement with the previous micromagnetic SANS data analysis of this sample (Michels et al., 2012;, we have set the ratio A H =A M ¼ 0:2. We also assumed that both spin-flip channels are equal, i.e. dAE þÀ =d ¼ dAE Àþ =d, a constraint that was already imposed during the spin-leakage correction. The overall qualitative agreement between experiment and theory (no free parameters) is evident, although the angular anisotropy of the data does not exhibit a large variation with field. Only at the smallest momentum transfers can one notice a change in the anisotropy with decreasing field (in particular in P À f? ), which is related to the emerging spin-misalignment scattering; compare e.g. scattering terms / j e M M y j 2 cos 4 and / CT yz sin cos in equations (66) and (69). We also note the existence of (seemingly isotropic) scattering contributions at small q < $ 0:1 nm À1 (especially at 1:27 T), which are probably due to large-scale structures that are not contained in the micromagnetic theory [compare Figs. 6(a) and 6(e) and Figs. 7(a) and 7(e)].
Due to the relatively large statistical noise in the twodimensional P AE f? maps we did not fit the experimental data directly to the theoretical expressions. Therefore, in the following, we consider one-dimensional experimental polarization data, which were obtained by averaging the twodimensional polarized SANS cross sections over AE8 along the vertical direction ( ¼ 90 ). These averages were used in equations (23a) and (23b) to obtain P AE f? ðqÞ. The resulting data in Fig. 8 were then fitted using the general equations (44a) (38)-(40)], is computed at each field using the materials parameters A and M 0 ; A is treated here as an additional adjustable parameter. For ðqÞ we used equation (63), and the DMI has been ignored in the data analysis (l D ¼ 0). Since the P AE f? differ only by the e N N e M M y and e N N e M M z interference terms, we have fitted the P AE f? ðqÞ data corresponding to the same field simultaneously. The applied field H 0 has been corrected for demagnetizing effects.
The fits in Fig. 8 (solid lines) provide a reasonable description of the experimental data. The obtained values for M and H are shown in Fig. 9; H ffi 6-15 nm is at all fields consistently of the order of the particle size, while M takes on larger values between about 22 and 65 nm. For the exchangestiffness constant, we obtain (from the four local fits) best-fit values in the range A = (4.8-9.7) Â 10 À12 J m À1 . These values agree very well with data in the literature Bersweiler et al., 2022).
Clearly, more experiments are needed in order to further scrutinize the predictions of the present micromagnetic theory for the uniaxial polarization analysis of bulk ferromagnets. In

Figure 9
Resulting best-fit values for the correlation lengths M and H (see inset). Lines are a guide to the eye. this respect, the development of computational tools to directly analyze the two-dimensional polarization maps using different form-factor and structure-factor expressions for e M M 2 z and e H H 2 p , and possibly the inclusion of a particle-size distribution function, would be desirable. Likewise, SANS measurements at a preferably saturating magnetic field are necessary to determine the nuclear SANS cross section, e.g. via a horizontal average of the non-spin-flip SANS cross section.

Summary and conclusions
We have provided a micromagnetic theory for the uniaxial polarization of the scattered neutrons of bulk ferromagnets, as it can be measured by means of the small-angle neutron scattering (SANS) method. The theoretical expressions contain the effects of an isotropic exchange interaction, the Dzyaloshinskii-Moriya interaction, magnetic anisotropy, magnetodipolar interaction and an external magnetic field. The theory has been employed to analyze experimental data on a soft magnetic nanocrystalline alloy; it may provide information on the magnetic interactions (exchange and DMI constants) and on the spatial structures of the magnetic anisotropy and magnetostatic fields. Given that uniaxial polarization analysis is becoming more and more available on SANS instruments worldwide and in view of the recent seminal progress made regarding several techniques which exploit the neutron polarization degree of freedom to characterize large-scale magnetic structures (SESANS, DFI, SEMSANS), we believe that the results of this paper open up a new avenue for magnetic neutron data analysis on mesoscopic magnetic systems. This is because the presented micromagnetic SANS framework forms the basis for all of these new and promising polarization encoding techniques, with the paper by Rekveldt et al. (2006) providing the relevant expressions that link the magnetization distribution to the final polarization and the projected correlation function.
Moreover, we note that dAE þÀ =d ¼ dAE Àþ =d for many polycrystalline bulk ferromagnets (Honecker et al., 2010). However, in our theoretical treatment we explicitly take into account the polarization dependence of the SANSPOL and spin-flip cross sections via the chiral function ðqÞ. This is relevant e.g. for systems where inversion symmetry is broken and the DMI is operative Quan et al., 2020).

APPENDIX B
Selected results for the polarization of the scattered neutrons of bulk ferromagnets In this appendix we provide some selected graphical representations for the dependency of the polarization of the scattered neutrons on the magnitude and orientation of the scattering vector, the applied magnetic field, the ratio of A M to A H , the ratio of nuclear to longitudinal magnetic scattering, and the DMI (via the exchange length l D ). Only results for k 0 ? H 0 are shown. The following materials parameters are used: A = 4.7 pJ m À1 ; 0 M 0 ¼ 1.64 T (l M ffi 2.1 nm); D = 2 mJ m À2 (l D ffi 1.9 nm) . Fig. 10 shows the two-dimensional final polarization P AE f? ðq y ; q z Þ and Fig. 11 depicts the corresponding 2 azimuthally averaged data P AE f? ðqÞ for different values of the applied magnetic field H i , ¼ ðqÞ [equation (63)], A H =A M ¼ 0:5, l D ¼ 0. The local extrema in P AE f? ðqÞ at small q are due to exp ðqÞ; setting ¼ constant results in smooth and continuously decaying functions. Figs. 12 and 13 display the polarization P AE ðqÞ for different ratios of A H =A M (Fig. 12) and for different (constant) values (Fig. 13) at a constant field of 0 H i ¼ 0.3 T and for l D ¼ 0. Including the DMI results in asymmetric P AE patterns at nonsaturating fields (see Fig. 14 Fig. 15. Here, peak-type features may appear in P AE , which might be detected in highly monodisperse particulate systems. Plot of P þ f? ðq y ; q z Þ (upper row) and P À f? ðq y ; q z Þ (lower row) for different applied magnetic fields H i (see insets). ¼ ðqÞ [equation (63)], A H =A M ¼ 0:5, l D ¼ 0.